Problem Statement

A nuclear fission source is contained within a spherical Inconel shell of outer radius $r_o=0\mathrm{.}05m$ and wall thickness $0\mathrm{.}01$ m $.$ It generates heat uniformly at a volumetric rate of $5{10}^5\frac{W}{m^3}.$ A primary coolant at a temperature of $T_{\text{i}\text{n}\text{f}}=500C$ flows around the shell. In order to increase heat transfer between the source and the coolant, $500$ triangular pin fins of base diameter $d_p=5mm$ and length $L_{{?}_{?}}(f)=35mm$ are attached to the shell. There is no contact resistance between the fins and the shell. The convective heat transfer coefficient between the coolant and the shell is $h=1000\frac{W}{m^2}-K.$ The thermal conductivity of the fission material, shell, and fin are respectively, $k_f=0\mathrm{.}2\frac{W}{m}-K,k_s=10\frac{W}{m}-K,$ and $k_{\text{f}\text{i}\text{n}}=100\frac{W}{m}-K.$

A$.$ Determine the temperature at the inner surface of the shell, $T_{si}$

B$.$ Determine the temperature at the center of the fission source, $T_{?}$

C$.$ Suggest a method by which $T_0$ can be decreased most significantly if the options are to increase by two fold the $($a$)$ thermal conductivity of the fission material without change to its heat generation rate, $($b$)$ fin density, or $($c$)$ convective heat transfer coefficient.

Note: Use the plot below to determine the efficiency of the triangular pin fin.